Choosing the Axes and the Scales

A basketball coach records how many hours seven players practice per week and how many free throws (out of 20) each one makes: | Player | A | B | C | D | E | F | G | |---|---|---|---|---|---|---|---| | Hours | 1 | 2 | 3 | 4 | 5 | 6 | 7 | | Free throws | 6 | 8 | 9 | 11 | 12 | 14 | 15 | Which variable goes on each axis, and what scales should you use?

Find the explanatory variable: Practice may influence the number of free throws, not the other way round = $x$ = hours of practice, $y$ = free throws made

Find the range of each variable: Hours: from $1$ to $7$. Free throws: from $6$ to $15$ = $x$ needs to reach at least $7$, $y$ at least $15$

Choose equal steps: $x$-axis: $0$ to $8$ in steps of $1$. $y$-axis: $0$ to $20$ in steps of $2$ (nobody can make more than 20 free throws, and $20 \div 2 = 10$ equal steps) = Both axes have equal spacing

Label the axes: $x$: Practice per week (hours). $y$: Free throws made (out of 20) = Axes are ready for plotting

Answer: Put **hours of practice on the $x$-axis** (0 to 8, steps of 1) and **free throws on the $y$-axis** (0 to 20, steps of 2), with a label and unit on each axis.

Plotting and Reading Points

Using the axes from Example 1, plot player C $(3, 9)$. Then use the finished plot to answer: how many players made **more than 10** free throws?

Move along the $x$-axis: Start at the origin and go right to $x = 3$ hours = You are above the 3

Move up to the $y$-value: Go straight up to $y = 9$ free throws and draw a dot = Point $(3, 9)$ is plotted

Plot every other row the same way: $(1, 6)$, $(2, 8)$, $(4, 11)$, $(5, 12)$, $(6, 14)$, $(7, 15)$ = 7 points, one per player

Read the plot: Points above the horizontal line $y = 10$: $(4, 11)$, $(5, 12)$, $(6, 14)$, $(7, 15)$ = $4$ players

Answer: Each player is one dot. **4 players** (D, E, F and G) made more than 10 free throws.

Describing a Linear Pattern

A café records the outside temperature and the number of hot chocolates sold: | Temperature (°C) | 0 | 5 | 10 | 15 | 20 | 25 | |---|---|---|---|---|---|---| | Hot chocolates | 48 | 40 | 33 | 25 | 18 | 10 | Describe the form, direction and strength of the scatter plot.

Check the direction: As temperature goes up, sales go down: $48 \to 40 \to 33 \to 25 \to 18 \to 10$ = Negative direction

Check the form: Every $5°C$ the sales drop by $8, 7, 8, 7, 8$: almost the same amount each time = Linear form

Check the strength: The drops are nearly equal, so the points lie very close to a straight line = Strong

Look for unusual features: No point is far from the others, no groups, no empty regions = No outliers, clusters or gaps

Answer: The scatter plot shows a **strong, negative, linear** association: on warmer days the café sells fewer hot chocolates.

Recognizing a Non-Linear Pattern

A driving school measures the braking distance of a car at different speeds: | Speed (km/h) | 20 | 40 | 60 | 80 | 100 | |---|---|---|---|---|---| | Braking distance (m) | 2 | 8 | 18 | 32 | 50 | Is the pattern linear or non-linear?

Check the direction: Faster speed, longer braking distance = Positive direction

Find the increases for equal steps of 20 km/h: $8 - 2 = 6$, $18 - 8 = 10$, $32 - 18 = 14$, $50 - 32 = 18$ = Increases: $6, 10, 14, 18$

Compare the increases: The increases keep growing, so the points bend upward instead of following a straight line = Non-linear (curved) form

Check the strength: Every point lies exactly on a smooth curve = Strong

Answer: The scatter plot shows a **strong, positive, non-linear** pattern: the curve gets steeper as the speed grows. Doubling the speed from 40 to 80 km/h makes the braking distance 4 times as long ($8 \to 32$ m).

Spotting Clusters, a Gap and an Outlier

A teacher plots homework time against quiz score (out of 10) for nine students: $(10, 4)$, $(15, 5)$, $(20, 5)$, $(25, 6)$, $(50, 8)$, $(55, 9)$, $(60, 9)$, $(65, 10)$, $(60, 2)$ Describe the unusual features of the scatter plot.

Look for groups of points: Four students worked 10 to 25 minutes and scored 4 to 6. Four students worked 50 to 65 minutes and scored 8 to 10 = Two clusters

Look for empty regions: No student worked between 25 and 50 minutes: $50 - 25 = 25$ minutes with no points = A gap on the $x$-axis

Look for a point that breaks the pattern: $(60, 2)$: 60 minutes of homework but only 2 points, while the others near 60 minutes scored 9 = $(60, 2)$ is an outlier

Describe the overall pattern: Without the outlier, more homework time goes with higher scores = Positive overall direction

Answer: There are **two clusters** (short and long homework time), a **gap** between 25 and 50 minutes and an **outlier** at $(60, 2)$. Overall the direction is positive. The outlier is worth a question: maybe that student was ill on the day of the quiz.

Mistake: Putting the response variable on the $x$-axis

Why: The axes have different jobs. If you plot free throws on the $x$-axis and practice on the $y$-axis, the graph suggests that free throws explain practice time.

Correct: Ask: which variable could influence the other? That one is the explanatory variable and goes on the $x$-axis.

Mistake: Using uneven steps on an axis (for example 0, 5, 10, 20, 50)

Why: Uneven spacing squeezes some points together and stretches others apart, so the pattern you see is false.

Correct: Use the same step all along each axis, for example 0, 10, 20, 30, 40, 50. The two axes may have different steps.

Mistake: Connecting the points with lines like a line graph

Why: The rows of the table are separate individuals, not moments in time. Joining them invents a path that does not exist.

Correct: Leave the points as separate dots. Describe the overall pattern in words (in the next lesson you will draw one line of best fit).

Mistake: Thinking a steep pattern is a strong pattern

Why: Steepness tells you how fast $y$ changes. Strength tells you how closely the points follow the pattern. A pattern with a gentle slope can be very strong, a steep one can be weak.

Correct: Judge strength by how tightly the points hug the line or curve, not by how steep it is.

Sports Science: Height and Arm Span

Coaches in swimming and basketball plot height against arm span, because a long arm span is an advantage in both sports.

For most people the arm span is almost equal to the height, so the points lie close to a straight line going up: a strong, positive, linear pattern.

Weather: Altitude and Temperature

Mountain guides and pilots know that the air gets colder the higher you go. A scatter plot of altitude against temperature shows how fast.

Weather balloons measure the temperature every few hundred meters on the way up. Plotted together, the readings fall almost on a straight line going down.

Business: Finding an Outlier

Shops plot the number of social media posts per week against the number of new customers to see if advertising works, and to spot weeks that do not fit.

A week with many posts but very few customers stands out as an outlier. It often has a reason: a holiday, a website problem or bad weather.

Q: What is the difference between a scatter plot and a line graph?

A: A line graph follows one quantity over time, so joining the points makes sense. A scatter plot compares two variables for many separate individuals, so the points stay unconnected.

Q: Do both axes have to start at 0?

A: No. If all heights are between 148 and 182 cm, starting the axis at 140 uses the space better. Just keep equal steps and make the starting value clear on the axis.

Q: What if neither variable explains the other?

A: Sometimes, like with height and arm span, neither one causes the other. Then you may choose either variable for the $x$-axis. Pick the one you would most likely use to estimate the other.

Q: Should I delete an outlier?

A: Not automatically. First check whether it is a mistake in the data. If it is a real value, keep it and mention it in your description: outliers often tell the most interesting story.